Reference book: 【KREYSZIG At AT Introductory Functional Analysis with Applications Complement of a set A 18, 609. Transpose of a matrix A 113 Space of bounded functions 228 A sequence space 70 Complex plane or the field of complex numbers 6, 51 Unitary-space 6 7 B[a, b] B(A) Space of bounded functions 11 BV a, b Space of functions of bounded variation 226 B(X, Y) B(x; r) Space of bounded linear operators 118 Open ball 18 B(x; r) Closed ball 18 C A sequence space 34 Co C C" C[a, b] Space of continuous functions C'[a, b] C(X, Y) D(T) Domain of an operator T 83 d(x, y) Distance from x to y 3 dim X δικ Kronecker delta 114 -(E) 1/ <(T) I inf L'[a, b] !" L(X, Y) M N(T) 0 " Space of continuously differentiable functions 110 Space of compact linear operators 411 Dimension of a space X 54 Spectral family 494 Norm of a bounded linear functional 104 Graph of an operator T 292 Identity operator 84 Infimum (greatest lower bound) 619 A function space 62 A sequence space 11 A sequence space 6 A space of linear operators 118 Annihilator of a set 148 Null space of an operator T 83 Zero operator 84 Empty set 609 1.3-1 Definition (Ball and sphere). Given a point xe X and a real number r>0, we define three types of sets: (a) B(xo; r) {xe X | d(x, x)0 there is a 8>0 such that" (see Fig. 6) d(Tx, Txo) 0, there exists a finite- rank operator F = B(X) such that ||T - F ||B(X) < ɛ. vazzanınægrænseqanto an¬nagrūovanno proveespace-
Reference book: 【KREYSZIG At AT Introductory Functional Analysis with Applications Complement of a set A 18, 609. Transpose of a matrix A 113 Space of bounded functions 228 A sequence space 70 Complex plane or the field of complex numbers 6, 51 Unitary-space 6 7 B[a, b] B(A) Space of bounded functions 11 BV a, b Space of functions of bounded variation 226 B(X, Y) B(x; r) Space of bounded linear operators 118 Open ball 18 B(x; r) Closed ball 18 C A sequence space 34 Co C C" C[a, b] Space of continuous functions C'[a, b] C(X, Y) D(T) Domain of an operator T 83 d(x, y) Distance from x to y 3 dim X δικ Kronecker delta 114 -(E) 1/ <(T) I inf L'[a, b] !" L(X, Y) M N(T) 0 " Space of continuously differentiable functions 110 Space of compact linear operators 411 Dimension of a space X 54 Spectral family 494 Norm of a bounded linear functional 104 Graph of an operator T 292 Identity operator 84 Infimum (greatest lower bound) 619 A function space 62 A sequence space 11 A sequence space 6 A space of linear operators 118 Annihilator of a set 148 Null space of an operator T 83 Zero operator 84 Empty set 609 1.3-1 Definition (Ball and sphere). Given a point xe X and a real number r>0, we define three types of sets: (a) B(xo; r) {xe X | d(x, x)0 there is a 8>0 such that" (see Fig. 6) d(Tx, Txo) 0, there exists a finite- rank operator F = B(X) such that ||T - F ||B(X) < ɛ. vazzanınægrænseqanto an¬nagrūovanno proveespace-
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.4: Definition Of Function
Problem 56E
Related questions
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![Reference book:
【KREYSZIG
At
AT
Introductory
Functional
Analysis with
Applications
Complement of a set A
18, 609.
Transpose of a matrix A 113
Space of bounded functions 228
A sequence space 70
Complex plane or the field of complex numbers 6, 51
Unitary-space 6
7
B[a, b]
B(A)
Space of bounded functions 11
BV a, b
Space of functions of bounded variation
226
B(X, Y)
B(x; r)
Space of bounded linear operators 118
Open ball 18
B(x; r)
Closed ball 18
C
A sequence space 34
Co
C
C"
C[a, b]
Space of continuous functions
C'[a, b]
C(X, Y)
D(T)
Domain of an operator T 83
d(x, y)
Distance from x to y 3
dim X
δικ
Kronecker delta 114
-(E)
1/
<(T)
I
inf
L'[a, b]
!"
L(X, Y)
M
N(T)
0
"
Space of continuously differentiable functions 110
Space of compact linear operators 411
Dimension of a space X 54
Spectral family 494
Norm of a bounded linear functional 104
Graph of an operator T 292
Identity operator 84
Infimum (greatest lower bound) 619
A function space 62
A sequence space 11
A sequence space 6
A space of linear operators 118
Annihilator of a set 148
Null space of an operator T 83
Zero operator 84
Empty set 609
1.3-1 Definition (Ball and sphere). Given a point xe X and a real
number r>0, we define three types of sets:
(a)
B(xo; r) {xe X | d(x, x)<r}
(Open ball)
(1) (b)
B(xo; r) = {x = X | d(x, xo)≤r}
(Closed ball)
(c)
S(xo; r) = {xe X | d(x, x) = r}
(Sphere)
In all three cases, xo is called the center and r the radius.
1.3-2 Definition (Open set, closed set). A subset M of a metric space
X is said to be open if it contains a ball about each of its points. A
subset K of X is said to be closed if its complement (in X) is open, that
is, K-X-K is open.
1.3-3 Definition (Continuous mapping). Let X-(X, d) and Y-(Y, ā)
be metric spaces. A mapping T: XY is said to be continuous at
a point xoX if for every e>0 there is a 8>0 such that" (see Fig. 6)
d(Tx, Txo)<E
for all x satisfying
d(x, xo)<8.
T is said to be continuous if it is continuous at every point of X. ■
1.3-4 Theorem (Continuous mapping). A mapping T of a metric
space X into a metric space Y is continuous if and only if the inverse
image of any open subset of Y is an open subset of X.
All the required definitions and theorems are attached, now
need solution to given questions, stop copy pasting anything, I
want fresh correct solutions, if you do not know the answers
Problem 6: Operator Norm and Compactness in Banach Spaces
Statement:
Let X be a Banach space, and let B(X) denote the space of bounded linear operators on X.
Consider the operator norm || . ||B(X).
Tasks:
1. Characterization of Compact Operators:
Prove that an operator T € B(X) is compact if and only if it maps the unit ball of X to a
relatively compact subset of X.
2. Operator Norm Estimation:
Given two bounded operators S, TЄ B(X), show that
||ST|| 5(x) ≤ ||S|| 5(x) · ||T||5(X) -
3. Approximation by Finite-Rank Operators:
Prove that if X is separable and T is compact, then for every € > 0, there exists a finite-
rank operator F = B(X) such that
||T - F ||B(X) < ɛ.
vazzanınægrænseqanto an¬nagrūovanno proveespace-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba9238e9-0b8a-4b88-943f-cd42df4cee1c%2F7bb75c29-bd83-4350-965a-116585ff859d%2Fxjubpg_processed.png&w=3840&q=75)
Transcribed Image Text:Reference book:
【KREYSZIG
At
AT
Introductory
Functional
Analysis with
Applications
Complement of a set A
18, 609.
Transpose of a matrix A 113
Space of bounded functions 228
A sequence space 70
Complex plane or the field of complex numbers 6, 51
Unitary-space 6
7
B[a, b]
B(A)
Space of bounded functions 11
BV a, b
Space of functions of bounded variation
226
B(X, Y)
B(x; r)
Space of bounded linear operators 118
Open ball 18
B(x; r)
Closed ball 18
C
A sequence space 34
Co
C
C"
C[a, b]
Space of continuous functions
C'[a, b]
C(X, Y)
D(T)
Domain of an operator T 83
d(x, y)
Distance from x to y 3
dim X
δικ
Kronecker delta 114
-(E)
1/
<(T)
I
inf
L'[a, b]
!"
L(X, Y)
M
N(T)
0
"
Space of continuously differentiable functions 110
Space of compact linear operators 411
Dimension of a space X 54
Spectral family 494
Norm of a bounded linear functional 104
Graph of an operator T 292
Identity operator 84
Infimum (greatest lower bound) 619
A function space 62
A sequence space 11
A sequence space 6
A space of linear operators 118
Annihilator of a set 148
Null space of an operator T 83
Zero operator 84
Empty set 609
1.3-1 Definition (Ball and sphere). Given a point xe X and a real
number r>0, we define three types of sets:
(a)
B(xo; r) {xe X | d(x, x)<r}
(Open ball)
(1) (b)
B(xo; r) = {x = X | d(x, xo)≤r}
(Closed ball)
(c)
S(xo; r) = {xe X | d(x, x) = r}
(Sphere)
In all three cases, xo is called the center and r the radius.
1.3-2 Definition (Open set, closed set). A subset M of a metric space
X is said to be open if it contains a ball about each of its points. A
subset K of X is said to be closed if its complement (in X) is open, that
is, K-X-K is open.
1.3-3 Definition (Continuous mapping). Let X-(X, d) and Y-(Y, ā)
be metric spaces. A mapping T: XY is said to be continuous at
a point xoX if for every e>0 there is a 8>0 such that" (see Fig. 6)
d(Tx, Txo)<E
for all x satisfying
d(x, xo)<8.
T is said to be continuous if it is continuous at every point of X. ■
1.3-4 Theorem (Continuous mapping). A mapping T of a metric
space X into a metric space Y is continuous if and only if the inverse
image of any open subset of Y is an open subset of X.
All the required definitions and theorems are attached, now
need solution to given questions, stop copy pasting anything, I
want fresh correct solutions, if you do not know the answers
Problem 6: Operator Norm and Compactness in Banach Spaces
Statement:
Let X be a Banach space, and let B(X) denote the space of bounded linear operators on X.
Consider the operator norm || . ||B(X).
Tasks:
1. Characterization of Compact Operators:
Prove that an operator T € B(X) is compact if and only if it maps the unit ball of X to a
relatively compact subset of X.
2. Operator Norm Estimation:
Given two bounded operators S, TЄ B(X), show that
||ST|| 5(x) ≤ ||S|| 5(x) · ||T||5(X) -
3. Approximation by Finite-Rank Operators:
Prove that if X is separable and T is compact, then for every € > 0, there exists a finite-
rank operator F = B(X) such that
||T - F ||B(X) < ɛ.
vazzanınægrænseqanto an¬nagrūovanno proveespace-
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